the muppet’s guide to: the structure and dynamics of solids 7. phase diagrams
TRANSCRIPT
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The Muppet’s Guide to:The Structure and Dynamics of Solids
7. Phase Diagrams
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Solid Solutions
Solute atoms create strain fields which can inhibit dislocations propagating in a material changing its properties
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Hume-Rothery Rules – Substitutional Solutions
1. The solute and solvent should be of a similar size. (<15% difference)
2. The crystal structures must match.
3. Both solute and solvent should have similar electronegativity
4. The valence of the solvent and solute metals should be similar.
Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)
Metals – Ni/Cu, Pd/Sn, Ag/Au, Mo/W
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Phase Equilibria – Example
CrystalStructure
electroneg r (nm)
K BCC 0.93 0.235
Na BCC 1.00 0.191
• Both have the same crystal structure (BCC) and have similar electronegativities but different atomic radii.
• Rules suggest that NO solid solution will form.
K-Na
• K and Na sodium are not miscible.
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Phase Equilibria – Example
CrystalStructure
electroneg r (nm)
Ni FCC 1.9 0.1246
Cu FCC 1.8 0.1278
• Both have the same crystal structure (FCC) and have similar electronegativities and atomic radii (W. Hume – Rothery rules) suggesting high mutual solubility.
Simple solution system (e.g., Ni-Cu solution)
• Ni and Cu are totally miscible in all proportions.
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Hume-Rothery Rules – Interstitial Solution
1. The solute must be smaller than the interstitial sites in the solvent lattice
2. Solute and Solvent should have similar electro-negativities
Rules to describe how an element might dissolve in a metal. Stable composition in equilibrium (thermodynamics)
Light elements – H,C, N and O.
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• Impurity atoms distort the lattice & generates stress.• Stress can produce a barrier to dislocation motion.
Modify Material Properties
• Smaller substitutional impurity
Impurity generates local stress at A and B that opposes dislocation motion to the right.
A
B
• Larger substitutional impurity
Impurity generates local stress at C and D that opposes dislocation motion to the right.
C
D
Increase material strength through substitution
(Callister: Materials Science and Engineering)
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Material PropertiesDislocations & plastic deformation• Cubic & hexagonal metals - plastic deformation by
plastic shear or slip where one plane of atoms slides over adjacent plane by defect motion (dislocations).
• If dislocations don't move, deformation doesn't occur!
Adapted from Fig. 7.1, Callister 7e.
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Edge Defect Motion
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Dislocations & Materials Classes
• Covalent Ceramics
(Si, diamond): Motion hard. -directional (angular) bonding
• Ionic Ceramics (NaCl):
Motion hard. -need to avoid ++ and - - neighbours.
+ + + +
+++
+ + + +
- - -
----
- - -
• Metals: Disl. motion easier.
-non-directional bonding -close-packed directions for slip.
electron cloud ion cores
++
++
++++++++ + + + + +
+++++++
(Callister: Materials Science and Engineering)
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Dislocation Motion• Dislocation moves along slip plane in slip direction
perpendicular to dislocation line• Slip direction same direction as Burgers vector
Edge dislocation
Screw dislocation
Adapted from Fig. 7.2, Callister 7e.
(Callister: Materials Science and Engineering)
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Pinning dislocations
• dislocations make metals easier to deform• to improve strength of metals, need to stop dislocation motion
trap with:- impurity atoms;- other dislocations (work hardening;
- grain boundaries.
atom trap
(Callister: Materials Science and Engineering)
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Modify Material Properties
• Grain boundaries are barriers to slip.• Barrier "strength" increases with Increasing angle of miss-orientation.• Smaller grain size: more barriers to slip.
Increase material strength through reducing Grain size
(Callister: Materials Science and Engineering)
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Solid Solutions
Solid state mixture of one or more solutes in a solvent
Crystal structure remains unchanged on addition of the solute to the solvent
Mixture remains in a homogenous phase
Generally composed on metals close in the periodic tableNi/Cu, Pb/Sn etc.
Otherwise compounds tend to formNaCl, Fe2O3 etc.
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• Components: The elements or compounds which are present in the mixture (e.g., Al and Cu)
• Phases: The physically and chemically distinct material regions that result (e.g., a and b).
Aluminum-CopperAlloy
Components and Phases
a (darker phase)
b (lighter phase)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Phase Diagrams
• A phase diagram is a graphical representation of the different phases present in a material.
• Commonly presented as a function of composition and temperature or pressure and temperature
Applies to elements, molecules etc. and can also be used to show magnetic, and ferroelectric behaviour (field vs. temperature) as well as structural information.
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Unary Phase DiagramsA pressure-temperature plot showing the different phases present in H2O.
Phase Boundaries
Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown
Crossing any line
results in a structural
phase transition
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Reading Unary Phase Diagrams
Melting Point (solid → liquid)
Boiling Point(liquid→ gas)
Sublimation (solid → gas)
As the pressure falls, the boiling point reduces, but the melting/freezing point remains reasonably constant.
Triple Point (solid + liquid + gas)
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Reading Unary Phase Diagrams
Melting Point: 0°C Boiling Point: 100°C
Melting Point: 2°C Boiling Point: 68°C
P=1atm
P=0.1atm
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Water Ice
http://images.jupiterimages.com/common/detail/13/41/23044113.jpg, http://www.homepages.ucl.ac.uk/~ucfbanf/ice_phase_diagram.jpg
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• When we combine two elements... what equilibrium state do we get?• In particular, if we specify... --a composition (e.g., wt.% Cu – wt.% Ni), and --a temperature (T )
then... How many phases do we get? What is the composition of each phase? How much of each phase do we get?
Binary Phase DiagramsPhase BPhase A
Nickel atomCopper atom
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Phase Equilibria: Solubility Limit– Solutions – solid solutions, single phase– Mixtures – more than one phase
• Solubility Limit: Max concentration for which only a single phase solution occurs.
Question: What is the solubility limit at 20°C?
Answer: 65 wt% sugar.
If Co < 65 wt% sugar: syrup
If Co > 65 wt% sugar: syrup + sugar.65
Sucrose/Water Phase Diagram
Pu
re
Su
gar
Tem
per
atu
re (
°C)
0 20 40 60 80 100Co =Composition (wt% sugar)
L (liquid solution
i.e., syrup)
Solubility Limit L
(liquid)
+ S
(solid sugar)20
40
60
80
100
Pu
re
Wat
er
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Salt-Water(ice)
http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif
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Ferroelectric Materials
Na½Ba½TiO3 T. Takenaka, K. Maruyama, and K. Sakata, Jpn. J. Appl. Phys. 30, 2236 (1991)
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B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic Press, London, 1971).
J. Rodel, W. Jo, K. T. P. Seifert, E. M. Anton, T. Granzow, and D. Damjanovic, J. Am. Ceram. Soc. 92, 1153 (2009).
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Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• PhaseDiagramfor Cu-Niat P=1 atm.
• 2 phases: L (liquid)
a (FCC solid solution)
• 3 phase fields: LL + aa
wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + aliquidus
solid
us
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Phase Diagrams• Indicate phases as function of T, Co, and P. • For this course: -binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• PhaseDiagramfor Cu-Niat P=1 atm.
wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + aliquidus
solid
us
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
Liquidus:Separates the liquid from the mixed L+ aphase
Solidus:Separates the mixed L+ a phase from the solid solution
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wt% Ni20 40 60 80 10001000
1100
1200
1300
1400
1500
1600T(°C)
L (liquid)
a (FCC solid solution)
L + a
liquidus
solid
us
Cu-Niphase
diagram
Number and types of phases• Rule 1: If we know T and Co, then we know: - the number and types of phases present.
• Examples:
A(1100°C, 60): 1 phase: a
B(1250°C, 35): 2 phases: L + a
B (
1250
°C,3
5) A(1100°C,60)
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + a
Cu-Ni system
Composition of phases• Rule 2: If we know T and Co, then we know: --the composition of each phase.
• Examples:TA
A
35Co
32CL
At TA = 1320°C:
Only Liquid (L) CL = Co ( = 35 wt% Ni)
At TB = 1250°C:
Both a and L CL = C liquidus ( = 32 wt% Ni here)
Ca = C solidus ( = 43 wt% Ni here)
At TD = 1190°C:
Only Solid ( a) Ca = Co ( = 35 wt% Ni)
Co = 35 wt% Ni
BTB
DTD
tie line
4Ca3
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
1200
1300
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35Co
L: 35wt%Ni
Cu-Nisystem
• Phase diagram: Cu-Ni system.
• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.
• Consider Co = 35 wt%Ni.
Cooling a Cu-Ni Binary - Composition
4635
4332
a: 43 wt% Ni
L: 32 wt% Ni
L: 24 wt% Ni
a: 36 wt% Ni
Ba: 46 wt% NiL: 35 wt% Ni
C
D
E
24 36
Figure adapted from Callister, Materials science and engineering, 7 th Ed. USE LEVER RULE
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• Tie line – connects the phases in equilibrium with each other - essentially an isotherm
The Lever Rule – Weight %
How much of each phase? Think of it as a lever
ML M
R S
RMSM L
L
L
LL
LL CC
CC
SR
RW
CC
CC
SR
S
MM
MW
00
wt% Ni
20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + aB
TB
tie line
CoC L Ca
SR
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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• Rule 3: If we know T and Co, then we know: --the amount of each phase (given in wt%).• Examples:
At T A : Only Liquid (L) W L = 100 wt%, W a = 0
At T D : Only Solid ( a) W L = 0, W a = 100 wt%
Co = 35 wt% Ni
Weight fractions of phases – ‘lever rule’
wt% Ni20
1200
1300
T(°C)
L (liquid)
a(solid)L + a
liquidus
solidus
30 40 50
L + a
Cu-Ni system
TA A
35Co
32C L
BTB
DTD
tie line
4Ca3
R S
= 27 wt%
43 3573 %
43 32wt
At T B : Both a and L
WL= SR + S
Wa= RR + S
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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wt% Ni20
120 0
130 0
30 40 50110 0
L (liquid)
a (solid)
L + a
L + a
T(°C)
A
35Co
L: 35wt%Ni
Cu-Nisystem
• Phase diagram: Cu-Ni system.• System is: --binary i.e., 2 components: Cu and Ni. --isomorphous i.e., complete solubility of one component in another; a phase field extends from 0 to 100 wt% Ni.• Consider Co = 35 wt%Ni.
Cooling a Cu-Ni Binary – wt. %
46344332
a: 27 wt%
L: 73 wt%
L: 8 wt%
a: 92 wt%
Ba: 8 wt% L: 92 wt%
C
D
E
24 36
Figure adapted from Callister, Materials science and engineering, 7 th Ed.
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Equilibrium cooling
• Multiple freezing sites– Polycrystalline materials– Not the same as a single crystal
• The compositions that freeze are a function of the temperature
• At equilibrium, the ‘first to freeze’ composition must adjust on further cooling by solid state diffusion
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Diffusion is not a flow
Our models of diffusion are based on a random walk approach and not a net flow
http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif
Concept behind mean free path in scattering phenomena - conductivity
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